On the multipoint distribution formulas of the parabolic Airy process

2025/07/28 by Liu, Zhipeng, Ortiz, Aaron · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2507.20487

Abstract

The parabolic Airy process is the Airy₂ process minus a parabola, initially defined by its finite-dimensional distributions, which are given by a Fredholm determinant formula with the extended Airy kernel. This process is also the one-time spatial marginal of the KPZ fixed point with the narrow wedge initial condition. There are two formulas for the space-time multipoint distribution of the KPZ fixed point with the narrow wedge initial condition obtained by arXiv:1906.01053 and arXiv:1907.09876. Especially, the equal-time case of arXiv:1907.09876 gives a different formula of the multipoint distribution of the parabolic Airy process. In this paper, we present a direct proof that this formula matches the one with the extended Airy kernel. Some byproducts in the proof include several new formulas for the parabolic Airy process, and a generalization of the Andreief's identity.

Citations

• An upper tail field of the KPZ fixed point
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Cited by

• Multipoint distributions of the KPZ fixed point with compactly supported initial conditions

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